Metropolis-coupled Markov chain Monte Carlo

Metropolis-coupled Markov chain Monte Carlo (or MCMCMC or (MC)3) is a variant of Markov chain Monte Carlo in which multiple chains are run in parallel, each with a different "temperature" (Geyer, 1991). Only the information from the cold chain is recorded. Periodically, trees between chains may be swapped.

"Some of these chains are 'heated' by raising the posterior probability to a power β. For example, if f (ψ|X) is the posterior probability density distribution of the phylogenetic parameters, then a heated version of the posterior distribution is f (ψ|X)β. Here, β(0 < β < 1) is the heat value of the chain. Heating a Markov chain increases the acceptance probability of new states. A heated chain tends to accept more states than a cold chain, allowing a heated chain to more readily cross valleys in the landscapes of trees." (Altekar et al., 2004).

(MC)3 can be used to empirically determine the posterior probability distribution of trees, branch lengths, and substitution parameters.

Temperature

A variable used in Metropolis-coupled Markov chain Monte Carlo. The temperature affects the likelihood of acceptance of a proposed tree and also affects the likelihood that two chains will accept a proposed tree swap.

Practice

Only one chain is sampled

The other chains are heated (i.e. they can take bigger steps)

Chains can swap states

Allows crossing of valleys

This process of linking heated chains is called "Metropolis-coupling"

The full-fledged Bayesian analysis is MCMCMC!

The heated chains use powers of the likelihood ratio in their acceptance ratio

We must only use the main chain to calculate posterior probabilities.

In reality, all these factors involve very large numbers. It's not uncommon to throw away thousands of trees as part of the burn-in and calculate the posterior from millions of trees.